MEN OF MATHEMATICS In sharp distinction to this is the other phrase which is reiter- ated over and over again in mathematical writing: 'There exists.' For example, some would say 'there exists a whole number greater than 2', or 'there exists a number less than n and greater than n — 2.' The use of such phraseology definitely commits its user to the creed which Kronecker held to be untenable, unless, of course, the 'existence1 is proved by a con- struction. The existence is not proved for the sets (as defined above) which appear in Cantor's theory. These two ways of speaking divide mathematicians into two types: the *we can' men believe (possibly subconsciously) that mathematics is a purely human invention; the * there exists' men believe that mathematics has an extra-human 'existence' of its own, and that *we' merely come upon the 'eternal truths' of mathematics in our journey through life, in much the same way that a man taking a walk in a city comes across a number of streets with whose planning he had nothing whatever to do. Theologians are 'exist' men; cautious sceptics for the most part 'we' men. 'There exist an infinity of even numbers, or of primes', say the advocates of extra-human 'existence'; 'produce them', say Kronecker and the 4we* men. That the distinction is not trivial can be seen from a famous instance of it in the New Testament. Christ asserted that the Father 'exists'; Philip demanded 'Show us the Father and it sufficeth us.' Cantor's theory is almost wholly on the 'existence' side. Is it possible that Cantor's passion for theology deter- mined his allegiance? If so, we shall have to explain why Kronecker, also a connoisseur of Christian theology, was the rabid *we' man that he was. As in all such questions ammunition for either side can be filched from any pocket. A striking and important instance of the 'existence' way of looking at the theory of sets is afforded by what is known as Zermelo's postulate (stated in 1904). 'For every set M whose elements are sets P (that is, M is a set of sets, or a class of classes), the sets P being non-empty and non-overlapping (no two contain things in common), there exists at least one set fi which contains precisely one element from each of the sets P which constitute M.' Comparison of this with the previously 632